Example — Prisoner's dilemma

The most famous game in the theory: two rational players, each with a dominant strategy, end up in an outcome worse for both than the cooperative one.

Example — Dominant strategy: the prisoner's dilemma (classic form)

Two suspects interrogated separately. Each can Tacere (T, stay silent) or Confessare (C, confess). If both stay silent: $1$ year in prison each. If both confess: $5$ years. If one confesses and the other stays silent: the one who confesses goes free ($0$), the one who stays silent gets $10$ years. Payoff = $-$years (we want to maximise):

G1 / G2	Tacere	Confessare
Tacere	$(-1;-1)$	$(-10; 0)$
Confessare	$(0;-10)$	$(-5;-5)$

Dominance analysis for G1.

• If G2 chooses T: G1 prefers C ($0 > -1$).
• If G2 chooses C: G1 prefers C ($-5 > -10$).

So C dominates T for G1. By symmetry C dominates T for G2 too.

Nash equilibrium: both confess (C, C) with payoff $(-5;-5)$. It is a pure-strategy equilibrium — but Pareto inefficient: the outcome (T, T) with payoff $(-1;-1)$ would have been better for both. The dilemma shows that individually rational equilibrium can lead to a suboptimal collective outcome.

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Topics: Probabilita
Concepts: Equilibrio di nash · Matrice di payoff · Strategia dominante
Methods: Equilibrio nash · Strategia dominante
Skills: Ragionare per casi